Matrix-Free Convex Optimization Modeling

S. Diamond and S. Boyd

Chapter in Optimization and Its Applications in Control and Data Sciences, Springer Optimization and Its Applications, B. Goldengorin, editor, 155:221–264, 2016.

Shorter version with title Convex Optimization with Abstract Linear Operators appeared in Proceedings International Conference on Computer Vision (ICCV), pages 675-683, December 2015.

We introduce a convex optimization modeling framework that transforms a convex optimization problem expressed in a form natural and convenient for the user into an equivalent cone program in a way that preserves fast linear transforms in the original problem. By representing linear functions in the transformation process not as matrices, but as graphs that encode composition of linear operators, we arrive at a matrix-free cone program, i.e., one whose data matrix is represented by a linear operator and its adjoint. This cone program can then be solved by a matrix-free cone solver. By combining the matrix-free modeling framework and cone solver, we obtain a general method for efficiently solving convex optimization problems involving fast linear transforms.